MathLabs

Problem 1

Prove that for every irrational real number a, there are irrational real numbers b and b' such that a+b and ab' are rational, while ab and a+b' are irrational.
Step 1 of 5: Choose b when a squared is irrational
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b=−aif a2∉Qb=-a\quad\text{if }a^2\notin\mathbb Q
Detailed analysis

If a squared is irrational, take b=-a. Then b is irrational, a+b=0 is rational, and ab=-a squared is irrational.